Anisotropic Shubin operators and eigenfunction expansions in Gelfand-Shilov spaces

We derive new results on the characterization of Gelfand--Shilov spaces $mathcal{S}^mu_ u (R^n)$, $mu, u >0$, $mu+ u geq 1$ by Gevrey estimates of the $L^2$ norms of iterates of $(m,k)$ anisotropic globally elliptic Shubin (or $Gamma$) type operators, $(-Delta)^{m/2} +| x |^k$ with $m,kin 2N$ being a model operator, and on the decay of the Fourier coefficients in the related eigenfunction expansions. Similar results are obtained for the spaces $Sigma^mu_ u (R^n)$, $mu, u >0$, $mu+ u > 1$, cf. eqref{GSdef}. In contrast to the symmetric case $mu = u$ and $k=m$ (classical Shubin operators) we encounter resonance type phenomena involving the ratio $kappa:=mu/ u$; namely we obtain a characterization of $mathcal{S}^mu_ u(R^n)$ and $Sigma^mu_ u(R^n)$ in the case $mu=kt/(k+m), u= mt/(k+m), t geq 1$, that is, when $kappa=k/m in Q$.